The focal distance of a point $(5, 5)$ on the parabola $x^2 - 2x - 4y + 5 = 0$ is

If $Q$ is the point on the parabola $y^2=4x$ that is nearest to the point $P(2,0)$, then $PQ=$

Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$,and let $Q$ and $Q^{\prime}$ be two distinct points on $E$ such that the lines $PQ$ and $PQ^{\prime}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) $TRUE$?
$(A)$ The triangle $PFQ$ is a right-angled triangle
$(B)$ The triangle $QPQ^{\prime}$ is a right-angled triangle
$(C)$ The distance between $P$ and $F$ is $5\sqrt{2}$
$(D)$ $F$ lies on the line joining $Q$ and $Q^{\prime}$

Find the equations of the tangent and normal to the parabola $y^{2}=4ax$ at the point $(at^{2}, 2at)$.

The focus of the parabola ${x^2} = 2x + 2y$ is

What is the focus of the parabola $y^2 - x - 2y + 2 = 0$?